No the allen wrench tool does not adjust the height of the coilover. All it does is tighten the perch so it does not slide, adjusting the height itself (although this is pretty impossible, as the weight of the car obviously holds everything in place).

Anyways, I've never seen the GC tool, as mine nor my friends' GC's came with it. I purchased a Gator Grip at Wal-Mart for 9.99, which is very similar to the tool used to tighten/loosen an oil filter. I tightened it around the gold perch, and had my friend hold it (but you could do it all yourself) while I turned the sleeve with a wrench. It worked great and only took a few minutes. I had to do this for all but my rear-right one which for some reason I could turn with my hand...

Good luck!